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Properties

Label 2-2940-35.4-c1-0-7
Degree $2$
Conductor $2940$
Sign $-0.652 - 0.758i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.133 + 2.23i)5-s + (0.499 + 0.866i)9-s + (2 − 3.46i)11-s + 6i·13-s + (1 − 1.99i)15-s + (−1.73 − i)17-s + (3 + 5.19i)19-s + (−1.73 + i)23-s + (−4.96 + 0.598i)25-s − 0.999i·27-s − 6·29-s + (1 − 1.73i)31-s + (−3.46 + 1.99i)33-s + (−3.46 + 2i)37-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.0599 + 0.998i)5-s + (0.166 + 0.288i)9-s + (0.603 − 1.04i)11-s + 1.66i·13-s + (0.258 − 0.516i)15-s + (−0.420 − 0.242i)17-s + (0.688 + 1.19i)19-s + (−0.361 + 0.208i)23-s + (−0.992 + 0.119i)25-s − 0.192i·27-s − 1.11·29-s + (0.179 − 0.311i)31-s + (−0.603 + 0.348i)33-s + (−0.569 + 0.328i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.652 - 0.758i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.652 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9672695905\)
\(L(\frac12)\) \(\approx\) \(0.9672695905\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.133 - 2.23i)T \)
7 \( 1 \)
good11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (-3.46 + 2i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.66 + 5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.242103620555177264701960390814, −8.105388164012065868849914166824, −7.41410659808962353164103449811, −6.61164082049660424229302461895, −6.17466968381565439307023756981, −5.40775090376249462953154215922, −4.13531291866282157797152923334, −3.56750328386962050365295764238, −2.35611428909548945271352882371, −1.39276657961476286445178085952, 0.34040283102022453510269112234, 1.45855051364898902012861049767, 2.72846656090350038755280325576, 3.97178550668080515087729535371, 4.59299321641505539406685428158, 5.43403797619505699826815701132, 5.90005658324623876628681226669, 7.11428712470146489509343523389, 7.60379872149318551507155340971, 8.700295490083000507309347200968

Graph of the $Z$-function along the critical line